This page was last edited on 12 December 2020, at 22:42. The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. 791.7 777.8] 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Theorem 1 (Frobenius). x��ZYo�6~�_�G5�fx�������d���yh{d[�ni"�q�_�U$����c�N���E�Y������(�4�����ٗ����i�Yvq�qbTV.���ɿ[�w��:��ȿo��{�XJ��7��}׷��jj?�o���UW��k�Mp��/���� /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 a 0x endobj 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 /Subtype/Type1 /FirstChar 33 An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 21 0 obj This is usually the method we use for complicated ordinary differential equations. \end{equation*}, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { m _ { j } + l } \left[ u ( z , \lambda ) ( \lambda - \lambda _ { j } ) ^ { m _ { j } } \right] = \end{equation*}, \begin{equation*} = \frac { ( m _ { j } + l ) ! } /Type/Font 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 3. In the former case there’s obviously only one Frobenius solution. >> Method of Frobenius Example First Solution Second Solution (Fails) What is the Method of Frobenius? n; y2(x) =xr2. 761.6 272 489.6] 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 /Type/Font >> /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 For instance, with r= 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 endobj 1062.5 826.4] /Subtype/Type1 << 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 This method enables one to compute a fundamental system of solutions for a holomorphic differential equation near a regular singular point (cf. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 9 0 obj We classify a point x The second solution can contain logarithmic terms in the higher powers starting with$( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$. /FontDescriptor 8 0 R Press (1989). The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation, $$\tag{a4} L _ { 0 } ( u ) = 0,$$, where the differential operator$L_0$is made from (a1) by retaining only the leading terms. endobj >> 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 << 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 /FirstChar 33 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /FontDescriptor 20 0 R /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. also Fuchsian equation). 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 In fact Frobenius method is just an extension from the power series method that you add an additional power that may not be an integer to each term in a power series or even add the log term for the assumptions of the solution form of the linear ODEs so that you can find all groups of the linearly independent solutions that in cases of cannot find all groups of the linearly independent solutions … as a recursion formula for$c_{j}$for all$j \geq 1$. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 Section 1.1 Frobenius Method In this section, we consider a method to find a general solution to a second order ODE about a singular point, written in either of the two equivalent forms below: $$x^2 y'' + xb(x)y' + c(x) y = 0\label{frobenius-standard-form1}\tag{1.1.1}$$ endobj Solve the hypergeometric equation around all singularities: 1. x ( 1 − x ) y ″ + { γ − ( 1 + α + β ) x } y ′ − α β y = 0 {\displaystyle x(1-x)y''+\left\{\gamma -(1+\alpha +\beta )x\right\}y'-\alpha \beta y=0} Example 3: x = 0 is an irregular point of the ﬂrst order equation Ly = x2y0 +y = 0 The solution of this ﬂrst order linear equation can be obtained by means of … For the case r= 1, we have a n = a n 1 5n+ 6 = ( 1)na 0 Yn k=1 (5j+ 1) 1; n= 1;2;:::; (36) and for r= 1 5, we have a n = a n 1 5n = ( 1)n 5nn! /FontDescriptor 35 0 R www.springer.com 4 Named after the German mathematician Ferdinand Georg Frobenius (1849 – 1917). Complications can arise if the generic assumption made above is not satisfied. 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 In particular, this can happen if the coe cients P(x) and Q(x) in the ODE y00+ P(x)y0+ Q(x)y = 0 fail to be de ned at a point x 0. Note that neither of the special cases below does exclude the simple generic case above. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 << /Name/F3 << Case I: Two equal roots If the indicial equation has two equal roots, γ1 = γ2, we have one solution of the form y1(t) = tγ1 ∞ ∑ n = 0cntn. Note that aFrobenius series is generally not power series. Hence, \begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i < j \ \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 The method looks simpler in the most common case of a differential operator, \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 The Set-Up The Calculations and Examples The Main Theorems Inserting the Series into the DE Getting the Coe cients Observations Coe cients We have, rst of all, F (r )=r (r 1 )+p 0 r +q 0 =0 ; the indicial equation. 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 27 0 obj 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Let y=Ún=0 ¥a xn+r. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 { N }$ ( 1 ) is given by y1 ( x ) =xr1 is given by y1 x! 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